Phy5645/Double pinhole experiment

Submitted by team 1

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(a) As stated in the problem, we assume that the denominators are approximately the same between the two waves. This is justified because the corrections are only on the order of $d/L\!$, and we are interested in the case where $d<<L\!$. We require that the numerators have the same phase, namely $kr_{+}-kr_{-}=2\pi n\!$. We expand the LHS with respect to $d\!$,

${\begin{aligned}k(r_{+}-r_{-})&\approx k\left\{\left({\sqrt {L^{2}+x^{2}+y^{2}}}+{\frac {yd}{2{\sqrt {L^{2}+x^{2}+y^{2}}}}}\right)\right.\\&-\left.\left({\sqrt {L^{2}+x^{2}+y^{2}}}-{\frac {yd}{2{\sqrt {L^{2}+x^{2}+y^{2}}}}}\right)\right\}\\&=k{\frac {dy}{\sqrt {L^{2}+x^{2}+y^{2}}}}\end{aligned}}$

Therefore,$k{\frac {dy}{\sqrt {L^{2}+x^{2}+y^{2}}}}=2\pi n\!$

and hence

${\frac {y}{\sqrt {L^{2}+x^{2}+y^{2}}}}=n{\frac {\lambda }{d}}\!$

(b) Let us work in units in which $k=1$; that is, we measure all lengths in units of $1/k$. We then set $d=20$ and $L=1000$. Here is the interference pattern. We first plot the pattern along the $y$ axis $(x=0)$:

(c) Using the same parameters as before, we now plot the pattern on the $xy$ plane:

(d) We use the same parameters as in (b) and (c). First, we plot the pattern along the $y$ axis $(x=0)$:

We now plot the pattern on the plane:

The main difference is the absence of the interference pattern.

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